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Theorem hbexgVD 38962
Description: Virtual deduction proof of hbexg 38592. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbexg 38592 is hbexgVD 38962 without virtual deductions and was automatically derived from hbexgVD 38962. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(𝜑 → ∀𝑥𝜑)   )
2:1: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥(𝜑 → ∀𝑥𝜑)   )
3:2: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (𝜑 → ∀𝑥𝜑)   )
4:3: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝜑 → ∀𝑥¬ 𝜑)   )
5:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦 𝑥(𝜑 → ∀𝑥𝜑))
6:: (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑦𝑥(𝜑 → ∀𝑥𝜑))
7:5: (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ 𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
8:5,6,7: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦 𝑥𝑦(𝜑 → ∀𝑥𝜑))
9:8,4: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝑥𝜑 → ∀𝑥¬ 𝜑)   )
10:9: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦𝜑 → ∀𝑥¬ 𝜑)   )
11:10: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 𝜑 → ∀𝑥¬ 𝜑)   )
12:11: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
13:12: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀ 𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
14:: (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥 𝑥𝑦(𝜑 → ∀𝑥𝜑))
15:13,14: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (∀𝑦¬ 𝜑 → ∀𝑥𝑦¬ 𝜑)   )
16:15: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 (¬ ∀𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
17:16: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶    𝑦¬ 𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
18:: (∃𝑦𝜑 ↔ ¬ ∀𝑦¬ 𝜑)
19:17,18: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥¬ ∀𝑦¬ 𝜑)   )
20:18: (∀𝑥𝑦𝜑 ↔ ∀𝑥¬ ∀𝑦¬ 𝜑)
21:19,20: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃ 𝑦𝜑 → ∀𝑥𝑦𝜑)   )
22:8,21: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
23:14,22: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
qed:23: (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥 𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
Assertion
Ref Expression
hbexgVD (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem hbexgVD
StepHypRef Expression
1 hba1 2149 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥𝑦(𝜑 → ∀𝑥𝜑))
2 hba1 2149 . . . . 5 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
3 alcom 2035 . . . . 5 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
43albii 1745 . . . . 5 (∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦𝑦𝑥(𝜑 → ∀𝑥𝜑))
52, 3, 43imtr4i 281 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥𝑦(𝜑 → ∀𝑥𝜑))
6 idn1 38610 . . . . . . . . . . . . . . . . 17 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(𝜑 → ∀𝑥𝜑)   )
7 ax-11 2032 . . . . . . . . . . . . . . . . 17 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦𝑥(𝜑 → ∀𝑥𝜑))
86, 7e1a 38672 . . . . . . . . . . . . . . . 16 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥(𝜑 → ∀𝑥𝜑)   )
9 sp 2051 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
108, 9e1a 38672 . . . . . . . . . . . . . . 15 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(𝜑 → ∀𝑥𝜑)   )
11 hbntal 38589 . . . . . . . . . . . . . . 15 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
1210, 11e1a 38672 . . . . . . . . . . . . . 14 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
135, 12gen11nv 38662 . . . . . . . . . . . . 13 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑)   )
14 ax-11 2032 . . . . . . . . . . . . 13 (∀𝑦𝑥𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1513, 14e1a 38672 . . . . . . . . . . . 12 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
16 sp 2051 . . . . . . . . . . . 12 (∀𝑥𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦𝜑 → ∀𝑥 ¬ 𝜑))
1715, 16e1a 38672 . . . . . . . . . . 11 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦𝜑 → ∀𝑥 ¬ 𝜑)   )
18 hbalg 38591 . . . . . . . . . . 11 (∀𝑦𝜑 → ∀𝑥 ¬ 𝜑) → ∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
1917, 18e1a 38672 . . . . . . . . . 10 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
20 sp 2051 . . . . . . . . . 10 (∀𝑦(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
2119, 20e1a 38672 . . . . . . . . 9 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
221, 21gen11nv 38662 . . . . . . . 8 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)   )
23 hbntal 38589 . . . . . . . 8 (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → ∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2422, 23e1a 38672 . . . . . . 7 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
25 sp 2051 . . . . . . 7 (∀𝑥(¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
2624, 25e1a 38672 . . . . . 6 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
27 df-ex 1703 . . . . . 6 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
28 imbi1 337 . . . . . . 7 ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) ↔ (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
2928biimprcd 240 . . . . . 6 ((¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3026, 27, 29e10 38739 . . . . 5 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)   )
3127albii 1745 . . . . 5 (∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
32 imbi2 338 . . . . . 6 ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∃𝑦𝜑 → ∀𝑥𝑦𝜑) ↔ (∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)))
3332biimprcd 240 . . . . 5 ((∃𝑦𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → ((∀𝑥𝑦𝜑 ↔ ∀𝑥 ¬ ∀𝑦 ¬ 𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑)))
3430, 31, 33e10 38739 . . . 4 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   (∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
355, 34gen11nv 38662 . . 3 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
361, 35gen11nv 38662 . 2 (   𝑥𝑦(𝜑 → ∀𝑥𝜑)   ▶   𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑)   )
3736in1 38607 1 (∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708  df-vd1 38606
This theorem is referenced by: (None)
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